We take the term randomly to mean that every possible

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We take the term “randomly” to mean that every possible partition is equally likely, so that the probability question can be reduced to one of counting
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According to our earlier discussion, there are = 16! / [4! 4! 4! 4! ] different partitions, and this is the size of the sample space. Let us now focus on the event that each group contains a graduate student. Generating an outcome with this property can be accomplished in two stages:
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a) Take the four graduate students and distribute them to the four groups; there are four choices for the group of the first graduate student, three choices for the second, two for the third. Thus, there is a total of 4! choices for this stage. (b) Take the remaining 12 undergraduate students and distribute them to the four groups (3 students in each). This can be done in = = 12! / 3! 3! 3! 3! different ways
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Summary of Counting Results Permutations of n objects: n ! • k -permutations of n objects: n ! / ( n − k )! Combinations of k out of n objects: nCk = n! / k !( n−k )!
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Partitions of n objects into r groups with the i th group having ni objects
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Example Anagrams. How many different letter sequences can be obtained by rearranging the letters in the word TATTOO? There are six positions to be filled by the available letters. Each rearrangement corresponds to a partition of the set of the six positions into a group of size 3 (the positions that get the letter T), a group of size 1 (the position that gets the letter A), and a group of size 2 (the positions that get the letter O). Thus, the desired number is 6! / (1! )(2!) (3!) = 60
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